Download Fundamentals of Linear Algebra

Willmar Public Schools
Curriculum Map
Subject Area
Mathematics
Course Name
Fundamentals of Linear Algebra
Date
6/10/09
Timeline
Content
Standards Addressed
Skills/Benchmarks
Essential Questions
Assessments
Sept
Sopris West
Transitional
Mathematics
Lesson 5&6
Read, write, compare,
classify and represent real
numbers, and use them to
solve problems in various
contexts.
Express approximations
of very large and very
small numbers using
scientific notation;
understand how
calculators display
numbers in scientific
notation. Multiply and
divide numbers expressed
in scientific notation,
express the answer in
scientific notation, using
the correct number of
significant digits when
physical measurements
are involved.
How do you write a very
large number using
multiplication and powers of
10?
Quizzes and tests.
Assignments completed.
For example:
(4.2 104 ) (8.25 103 ) 3.465 108
, but if these numbers represent
physical measurements, the
answer should be expressed as
3.5 108 because the first
factor, 4.2 104 , only has two
significant digits.
Lesson 9
Express approximations
of very large and very
small numbers using
scientific notation;
understand how
calculators display
How do you write a very
large number using
multiplication and powers of
10?
Quizzes and tests.
Assignments completed.
Willmar Public Schools
Curriculum Map
numbers in scientific
notation. Multiply and
divide numbers expressed
in scientific notation,
express the answer in
scientific notation, using
the correct number of
significant digits when
physical measurements
are involved.
For example:
(4.2 104 ) (8.25 103 ) 3.465 108
, but if these numbers represent
physical measurements, the
answer should be expressed as
3.5 108 because the first
factor, 4.2 104 , only has two
significant digits.
Lesson 11
Classify real numbers as
rational or irrational.
Know that when a square
root of a positive integer
is not an integer, then it is
irrational. Know that the
sum of a rational number
and an irrational number
is irrational, and the
product of a non-zero
rational number and an
irrational number is
irrational.
For example: Classify the
following numbers as whole
numbers, integers, rational
numbers, irrational numbers,
What do rational and
irrational numbers look like?
Quizzes and tests.
Assignments completed.
Willmar Public Schools
Curriculum Map
recognizing that some numbers
belong in more than one
category: 6 , 3 , 3.6 , ,
3
4,
Lesson 11
6
2
10 , 6.7 .
Compare real numbers;
locate real numbers on a
number line. Identify the
square root of a positive
integer as an integer, or if
it is not an integer, locate
it as a real number
between two consecutive
positive integers.
What is the location of
rational numbers on a
number line?
Quizzes and tests.
Assignments completed.
How do you round off
rational/irrational numbers?
Quizzes and tests.
Assignments completed.
For example: Put the following
numbers in order from smallest
to largest: 2, 3 , 4, 6.8,
37 .
Another example: 68 is an
irrational number between 8
and 9.
Oct
Sopris West
Transitional
Mathematics
Lesson 19
Read, write, compare,
classify and represent real
numbers, and use them to
solve problems in various
contexts.
Determine rational
approximations for
solutions to problems
involving real numbers.
For example: A calculator can
be used to determine that 7 is
approximately 2.65.
Another example: To check that
1 5 is slightly bigger than 2 ,
12
do the calculation
15
12
2
17
12
2
289
144
2 1
144
.
Another example: Knowing
that 10 is between 3 and 4,
Willmar Public Schools
Curriculum Map
try squaring numbers like 3.5,
3.3, 3.1 to determine that 3.1 is
a reasonable rational
approximation of 10 .
Lesson 21
Read, write, compare,
classify and represent real
numbers, and use them to
solve problems in various
contexts.
Know and apply the
properties of positive and
negative integer
exponents to generate
equivalent numerical
expressions.
What do exponents mean?
Quizzes and tests.
Assignments completed.
What happens to an
algebraic equation when the
value of x changes?
Quizzes and tests.
Assignments completed.
For example:
32 3
Nov
Sopris West
Transitional
Mathematics
Lesson 27
Understand the concept of
function in real-world and
mathematical situations,
and distinguish between
linear and nonlinear
functions.
5
3
3
1
3
3
1
27
.
Understand that a
function is a relationship
between an independent
variable and a dependent
variable in which the
value of the independent
variable determines the
value of the dependent
variable. Use functional
notation, such as f(x), to
represent such
relationships.
For example: The relationship
between the area of a square
and the side length can be
expressed as f ( x) x2 . In this
case, f (5) 25 , which
represents the fact that a square
of side length 5 units has area
25 units squared.
Willmar Public Schools
Curriculum Map
Dec
Jan
Sopris West
Transitional
Mathematics
Lesson 30
Sopris West
Transitional
Mathematics
Lesson 40-43
Lesson 45&46
Represent real-world and
mathematical situations
using equations and
inequalities involving linear
expressions. Solve
equations and inequalities
symbolically and
graphically. Interpret
solutions in the original
context.
Use linear equations to
represent situations
involving a constant rate
of change, including
proportional and nonproportional
relationships.
Represent real-world and
mathematical situations
using equations and
inequalities involving linear
expressions. Solve
equations and inequalities
symbolically and
graphically. Interpret
solutions in the original
context.
Use linear inequalities to
represent relationships in
various contexts.
How does a proportion help
you find an unknown
quantity?
Quizzes and tests.
Assignments completed.
What does < and > mean?
Quizzes and tests.
Assignments completed.
What is a proportion and
how does it relate to rate of
change?
Quizzes and tests.
Assignments completed.
For example: For a cylinder
with fixed radius of length 5,
the surface area A = 2π(5)h +
2π(5)2 = 10πh + 50π, is a linear
function of the height h, but the
surface area is not proportional
to the height.
For example: A gas station
charges $0.10 less per gallon of
gasoline if a customer also gets
a car wash. Without the car
wash, gas costs $2.79 per
gallon. The car wash is $8.95.
What are the possible amounts
(in gallons) of gasoline that you
can buy if you also get a car
wash and can spend at most
$35?
Use linear equations to
represent situations
involving a constant rate
of change, including
proportional and nonproportional
relationships.
For example: For a cylinder
with fixed radius of length 5,
Willmar Public Schools
Curriculum Map
the surface area A = 2π(5)h +
2π(5)2 = 10πh + 50π, is a linear
function of the height h, but the
surface area is not proportional
to the height.
Feb
Sopris West
Transitional
Mathematics
Lesson 52
Interpret data using
scatterplots and
approximate lines of best
fit. Use lines of best fit to
draw conclusions about
data.
Mar
Sopris West
Transitional
Mathematics
Lesson 58, 61-63
Understand the concept of
function in real-world and
mathematical situations,
and distinguish between
linear and nonlinear
functions.
Collect, display and
interpret data using
scatterplots. Use the
shape of the scatterplot to
informally estimate a line
of best fit and determine
an equation for the line.
Use appropriate titles,
labels and units. Know
how to use graphing
technology to display
scatterplots and
corresponding lines of
best fit.
Understand that a
function is a relationship
between an independent
variable and a dependent
variable in which the
value of the independent
variable determines the
value of the dependent
variable. Use functional
notation, such as f(x), to
represent such
relationships.
For example: The relationship
between the area of a square
and the side length can be
expressed as f ( x) x2 . In this
case, f (5) 25 , which
represents the fact that a square
Studying a scatterplot what
is the relationship between
the x and y values?
Quizzes and tests.
Assignments completed.
How do changes in your x
values change your resulting
y values?
Quizzes and tests.
Assignments completed.
Willmar Public Schools
Curriculum Map
of side length 5 units has area
25 units squared.
Apr
Sopris West
Transitional
Mathematics
Lesson 66 &67
Solve problems involving
right triangles using the
Pythagorean Theorem and
its converse.
Determine the distance
between two points on a
horizontal or vertical line
in a coordinate system.
Use the Pythagorean
Theorem to find the
distance between any two
points in a coordinate
system.
How many spaces are
between Point A and Point
B?
Quizzes and tests.
Assignments completed.
Lesson 68
Understand the concept of
function in real-world and
mathematical situations,
and distinguish between
linear and nonlinear
functions.
Understand that an
arithmetic sequence is a
linear function that can be
expressed in the
form f (x) mx b , where
x = 0, 1, 2, 3,….
What happens in a table
where the x=0,1,2,3,4…?
Quizzes and tests.
Assignments completed.
What happens to y when x is
changed?
Quizzes and tests.
Assignments completed.
For example: The arithmetic
sequence 3, 7, 11, 15, …, can
be expressed as f(x) = 4x + 3.
May/
June
Sopris West
Transitional
Mathematics
Lesson 68&70
Understand the concept of
function in real-world and
mathematical situations,
and distinguish between
linear and nonlinear
functions.
Use linear functions to
represent relationships in
which changing the input
variable by some amount
leads to a change in the
output variable that is a
constant times that
amount.
For example: Uncle Jim gave
Emily $50 on the day she was
born and $25 on each birthday
after that. The
function f ( x) 50 25 x represe
nts the amount of money Jim
has given after x years. The rate
Willmar Public Schools
Curriculum Map
of change is $25 per year.
Lesson 73
Understand the concept of
function in real-world and
mathematical situations,
and distinguish between
linear and nonlinear
functions.
Understand that a
function is linear if it can
be expressed in the
form f (x) mx b or if its
graph is a straight line.
What shape is the equation
y=mx+b when it is graphed?
Quizzes and tests.
Assignments completed.
Given an equation, graph the
Represent linear
line. What does the line
functions with tables,
represent?
verbal descriptions,
symbols, equations and
graphs; translate from one
representation to another.
Quizzes and tests.
Assignments completed.
Which number represents
the slope and the intercept in
the equation?
Quizzes and tests.
Assignments completed.
For example: The
function f ( x) x2 is not a
linear function because its
graph contains the points (1,1),
(-1,1) and (0,0), which are not
on a straight line.
Lesson 68-74
Lesson 73
Recognize linear functions
in real-world and
mathematical situations;
represent linear functions
and other functions with
tables, verbal descriptions,
symbols and graphs; solve
problems involving these
functions and explain
results in the original
context.
Identify graphical
properties of linear
functions including
slopes and intercepts.
Know that the slope
equals the rate of change,
and that the y-intercept is
zero when the function
Willmar Public Schools
Curriculum Map
represents a proportional
relationship.
Lesson 72
Recognize linear functions
in real-world and
mathematical situations;
represent linear functions
and other functions with
tables, verbal descriptions,
symbols and graphs; solve
problems involving these
functions and explain
results in the original
context.
Lesson 74
Identify how coefficient
changes in the equation f
(x) = mx + b affect the
graphs of linear
functions. Know how to
use graphing technology
to examine these effects.
If you change the slope,
what happens to the line?
Quizzes and tests.
Assignments completed.
Represent arithmetic
sequences using
equations, tables, graphs
and verbal descriptions,
and use them to solve
problems.
What does the y value mean
when x is given in a table or
graph?
Quizzes and tests.
Assignments completed.
What are the associative and
commutative properties?
Quizzes and tests.
Assignments completed.
For example: If a girl starts
with $100 in savings and adds
$10 at the end of each month,
she will have 100 + 10x dollars
after x months.
Lesson 80&81
Generate equivalent
numerical and algebraic
expressions and use
algebraic properties to
evaluate expressions.
Justify steps in generating
equivalent expressions by
identifying the properties
used, including the
properties of algebra.
Properties include the
associative, commutative
and distributive laws, and
the order of operations,
including grouping
Willmar Public Schools
Curriculum Map
symbols.
Lesson 72
Represent real-world and
mathematical situations
using equations and
inequalities involving linear
expressions. Solve
equations and inequalities
symbolically and
graphically. Interpret
solutions in the original
context.
Solve multi-step
equations in one variable.
Solve for one variable in
a multi-variable equation
in terms of the other
variables. Justify the
steps by identifying the
properties of equalities
used.
How do you use the order of
operations to solve
equations?
Quizzes and tests.
Assignments completed.
How do you put a known
slope and intercept into an
equation?
Quizzes and tests.
Assignments completed.
What does it mean when two
lines intersect?
Quizzes and tests.
Assignments completed.
For example: The equation 10x
+ 17 = 3x can be changed to 7x
+ 17 = 0, and then to 7x = -17
by adding/subtracting the same
quantities to both sides. These
changes do not change the
solution of the equation.
Another example: Using the
formula for the perimeter of a
rectangle, solve for the base in
terms of the height and
perimeter.
Lesson 73
Express linear equations
in slope-intercept, pointslope and standard forms,
and convert between
these forms. Given
sufficient information,
find an equation of a line.
For example: Determine an
equation of the line through the
points (-1,6) and (2/3, -3/4).
Lesson 72-74
Represent real-world and
mathematical situations
Represent relationships in
various contexts using
Willmar Public Schools
Curriculum Map
using equations and
inequalities involving linear
expressions. Solve
equations and inequalities
symbolically and
graphically. Interpret
solutions in the original
context.
Lesson 72&73
Lesson 73
Solve problems involving
parallel and perpendicular
lines on a coordinate
systems of linear
equations. Solve systems
of linear equations in two
variables symbolically,
graphically and
numerically.
For example: Marty's cell
phone company charges $15
per month plus $0.04 per
minute for each call. Jeannine's
company charges $0.25 per
minute. Use a system of
equations to determine the
advantages of each plan based
on the number of minutes used.
Understand that a system
of linear equations may
have no solution, one
solution, or an infinite
number of solutions.
Relate the number of
solutions to pairs of lines
that are intersecting,
parallel or identical.
Check whether a pair of
numbers satisfies a
system of two linear
equations in two
unknowns by substituting
the numbers into both
equations.
What does it mean for the
solution when two lines
intersect? Are parallel?
Quizzes and tests.
Assignments completed.
Understand and apply the
relationships between the
slopes of parallel lines
and between the slopes of
How can you determine that
two lines are parallel by
looking at their equations?
Quizzes and tests.
Assignments completed.
Willmar Public Schools
Curriculum Map
system.
Lesson 75
perpendicular lines.
Dynamic graphing
software may be used to
examine these
relationships.
Given a line on a
coordinate system and the
coordinates of a point not
on the line, find lines
through that point that are
parallel and perpendicular
to the given line,
symbolically and
graphically.
Content -- big ideas, broad topics, major subcategories and underlying concepts
Standards Addressed -- state and/or local standards
Skills/Benchmarks -- tells what the student will be able to do as a result of instruction
Essential Questions -- what overarching questions will guide instruction and produce higher levels of thinking?
Assessments -- evidence that the student understands the concepts, demonstration of skills
Given a line on a graph, how
do you draw a line parallel
to the line through a given
point?
Quizzes and tests.
Assignments completed.